Dynamic programming principle and Hamilton-Jacobi-Bellman equation under nonlinear expectation

TL;DR

This paper investigates a stochastic recursive control problem under nonlinear $ ilde{G}$-expectation, establishing a comparison theorem, a dynamic programming principle, and proving the value function as the unique viscosity solution of a nonlinear Hamilton-Jacobi-Bellman equation.

Contribution

It introduces a new approach to derive the dynamic programming principle for control problems under $ ilde{G}$-expectation and links the value function to a fully nonlinear HJB equation.

Findings

Comparison theorem for $ ilde{G}$-BSDEs established

Dynamic programming principle proved under nonlinear expectation

Value function characterized as unique viscosity solution

Abstract

In this paper, we study a stochastic recursive optimal control problem in which the value functional is defined by the solution of a backward stochastic differential equation (BSDE) under $\tilde{G}$-expectation. Under standard assumptions, we establish the comparison theorem for this kind of BSDE and give a novel and simple method to obtain the dynamic programming principle. Finally, we prove that the value function is the unique viscosity solution of a type of fully nonlinear HJB equation.